Thanks to everyone for your responses! I never expected to get this many
different ideas, and I guess I'll need to decide which ones to try first.

I've never seen or used Gorilla Glue, but I think I might try that first and
see what happens. If that doesn't seem to work out, I'll probably try the
wedge idea or something similar to make sure the mechanical connection
really solid with no gaps first before re-gluing.